\(\int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} (a+b x^2)} \, dx\) [1]

Optimal result

Integrand size = 37, antiderivative size = 230 \[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {\left (\sqrt {b} B C-\sqrt {-a} (A+B D)\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {b} \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e}}-\frac {\left (\sqrt {b} B C+\sqrt {-a} (A+B D)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {b} \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e}} \] Output:

-(b^(1/2)*B*C-(-a)^(1/2)*(B*D+A))*arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e 
*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(3/4)/b^(1/2)/(b^(1/2)*c- 
(-a)^(1/2)*d)^(1/2)/e^(1/2)-(b^(1/2)*B*C+(-a)^(1/2)*(B*D+A))*arctanh((b^(1 
/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(- 
a)^(3/4)/b^(1/2)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)/e^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.92 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.13 \[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\sqrt {x} \text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {B C d^3 \log (x)-2 B C d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )+4 A c d \log (x) \text {$\#$1}^2-3 B C d^2 \log (x) \text {$\#$1}^2+4 B c d D \log (x) \text {$\#$1}^2-8 A c d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+6 B C d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2-8 B c d D \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2-4 A c \log (x) \text {$\#$1}^4+3 B C d \log (x) \text {$\#$1}^4-4 B c D \log (x) \text {$\#$1}^4+8 A c \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-6 B C d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+8 B c D \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-B C \log (x) \text {$\#$1}^6+2 B C \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{a d^3 \text {$\#$1}-8 b c^2 \text {$\#$1}^3-3 a d^2 \text {$\#$1}^3+3 a d \text {$\#$1}^5-a \text {$\#$1}^7}\&\right ]}{4 \sqrt {e x}} \] Input:

Integrate[(B*C + (A + B*D)*x)/(Sqrt[e*x]*Sqrt[c + d*x]*(a + b*x^2)),x]
 

Output:

(Sqrt[x]*RootSum[a*d^4 - 4*a*d^3*#1^2 + 16*b*c^2*#1^4 + 6*a*d^2*#1^4 - 4*a 
*d*#1^6 + a*#1^8 & , (B*C*d^3*Log[x] - 2*B*C*d^3*Log[-Sqrt[c] + Sqrt[c + d 
*x] - Sqrt[x]*#1] + 4*A*c*d*Log[x]*#1^2 - 3*B*C*d^2*Log[x]*#1^2 + 4*B*c*d* 
D*Log[x]*#1^2 - 8*A*c*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 
6*B*C*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 - 8*B*c*d*D*Log[ 
-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 - 4*A*c*Log[x]*#1^4 + 3*B*C*d* 
Log[x]*#1^4 - 4*B*c*D*Log[x]*#1^4 + 8*A*c*Log[-Sqrt[c] + Sqrt[c + d*x] - S 
qrt[x]*#1]*#1^4 - 6*B*C*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 
+ 8*B*c*D*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - B*C*Log[x]*#1^ 
6 + 2*B*C*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6)/(a*d^3*#1 - 8*b 
*c^2*#1^3 - 3*a*d^2*#1^3 + 3*a*d*#1^5 - a*#1^7) & ])/(4*Sqrt[e*x])
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2353, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x (A+B D)+B C}{\sqrt {e x} \left (a+b x^2\right ) \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2353

\(\displaystyle \int \left (\frac {\sqrt {-a} B C-\frac {a (A+B D)}{\sqrt {b}}}{2 a \sqrt {e x} \left (\sqrt {-a}-\sqrt {b} x\right ) \sqrt {c+d x}}+\frac {\frac {a (A+B D)}{\sqrt {b}}+\sqrt {-a} B C}{2 a \sqrt {e x} \left (\sqrt {-a}+\sqrt {b} x\right ) \sqrt {c+d x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (B C-\frac {\sqrt {-a} (A+B D)}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (\frac {\sqrt {-a} (A+B D)}{\sqrt {b}}+B C\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}\)

Input:

Int[(B*C + (A + B*D)*x)/(Sqrt[e*x]*Sqrt[c + d*x]*(a + b*x^2)),x]
 

Output:

-(((B*C - (Sqrt[-a]*(A + B*D))/Sqrt[b])*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]* 
d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(3/4)*Sqrt[Sqrt[b 
]*c - Sqrt[-a]*d]*Sqrt[e])) - ((B*C + (Sqrt[-a]*(A + B*D))/Sqrt[b])*ArcTan 
h[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d* 
x])])/((-a)^(3/4)*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2353

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer 
Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1351\) vs. \(2(170)=340\).

Time = 0.29 (sec) , antiderivative size = 1352, normalized size of antiderivative = 5.88

method	result	size
default	\(\text {Expression too large to display}\)	\(1352\)

Input:

int((B*C+(B*D+A)*x)/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x,method=_RETURNVE 
RBOSE)
 

Output:

-1/2*(d*x+c)^(1/2)*x*e*(-B*C*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*(e*(c*(-a* 
b)^(1/2)-d*a)/b)^(1/2)*(x*e*(d*x+c))^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b) 
^(1/2)))*a*d^2*(-a*b)^(1/2)*(-e*(c*(-a*b)^(1/2)+d*a)/b)^(1/2)-B*C*ln((2*(- 
a*b)^(1/2)*d*e*x+b*c*e*x+2*(e*(c*(-a*b)^(1/2)-d*a)/b)^(1/2)*(x*e*(d*x+c))^ 
(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*b*c^2*(-a*b)^(1/2)*(-e*(c*(- 
a*b)^(1/2)+d*a)/b)^(1/2)+B*C*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*(-e*(c*(- 
a*b)^(1/2)+d*a)/b)^(1/2)*(x*e*(d*x+c))^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a* 
b)^(1/2)))*a*d^2*(e*(c*(-a*b)^(1/2)-d*a)/b)^(1/2)*(-a*b)^(1/2)+B*C*ln((-2* 
(-a*b)^(1/2)*d*e*x+b*c*e*x+2*(-e*(c*(-a*b)^(1/2)+d*a)/b)^(1/2)*(x*e*(d*x+c 
))^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*b*c^2*(e*(c*(-a*b)^(1/2)- 
d*a)/b)^(1/2)*(-a*b)^(1/2)+B*D*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*(e*(c*(- 
a*b)^(1/2)-d*a)/b)^(1/2)*(x*e*(d*x+c))^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a* 
b)^(1/2)))*a^2*d^2*(-e*(c*(-a*b)^(1/2)+d*a)/b)^(1/2)+B*D*ln((2*(-a*b)^(1/2 
)*d*e*x+b*c*e*x+2*(e*(c*(-a*b)^(1/2)-d*a)/b)^(1/2)*(x*e*(d*x+c))^(1/2)*b+c 
*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*a*b*c^2*(-e*(c*(-a*b)^(1/2)+d*a)/b)^( 
1/2)+B*D*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*(-e*(c*(-a*b)^(1/2)+d*a)/b)^( 
1/2)*(x*e*(d*x+c))^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*a^2*d^2*( 
e*(c*(-a*b)^(1/2)-d*a)/b)^(1/2)+B*D*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*(- 
e*(c*(-a*b)^(1/2)+d*a)/b)^(1/2)*(x*e*(d*x+c))^(1/2)*b-c*e*(-a*b)^(1/2))/(b 
*x+(-a*b)^(1/2)))*a*b*c^2*(e*(c*(-a*b)^(1/2)-d*a)/b)^(1/2)+A*ln((2*(-a*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4961 vs. \(2 (170) = 340\).

Time = 0.51 (sec) , antiderivative size = 4961, normalized size of antiderivative = 21.57 \[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((B*C+(B*D+A)*x)/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm 
="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int \frac {A x + B C + B D x}{\sqrt {e x} \left (a + b x^{2}\right ) \sqrt {c + d x}}\, dx \] Input:

integrate((B*C+(B*D+A)*x)/(e*x)**(1/2)/(d*x+c)**(1/2)/(b*x**2+a),x)
                                                                                    
                                                                                    
 

Output:

Integral((A*x + B*C + B*D*x)/(sqrt(e*x)*(a + b*x**2)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int { \frac {B C + {\left (B D + A\right )} x}{{\left (b x^{2} + a\right )} \sqrt {d x + c} \sqrt {e x}} \,d x } \] Input:

integrate((B*C+(B*D+A)*x)/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm 
="maxima")
 

Output:

integrate((B*C + (B*D + A)*x)/((b*x^2 + a)*sqrt(d*x + c)*sqrt(e*x)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*C+(B*D+A)*x)/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm 
="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int \frac {x\,\left (A+B\,D\right )+B\,C}{\sqrt {e\,x}\,\left (b\,x^2+a\right )\,\sqrt {c+d\,x}} \,d x \] Input:

int((x*(A + B*D) + B*C)/((e*x)^(1/2)*(a + b*x^2)*(c + d*x)^(1/2)),x)
 

Output:

int((x*(A + B*D) + B*C)/((e*x)^(1/2)*(a + b*x^2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {B C+(A+B D) x}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\left (\int \frac {x}{\sqrt {x}\, \sqrt {d x +c}\, a +\sqrt {x}\, \sqrt {d x +c}\, b \,x^{2}}d x \right ) a +\left (\int \frac {x}{\sqrt {x}\, \sqrt {d x +c}\, a +\sqrt {x}\, \sqrt {d x +c}\, b \,x^{2}}d x \right ) b d +\left (\int \frac {1}{\sqrt {x}\, \sqrt {d x +c}\, a +\sqrt {x}\, \sqrt {d x +c}\, b \,x^{2}}d x \right ) b c}{\sqrt {e}} \] Input:

int((B*C+(B*D+A)*x)/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x)
 

Output:

(int(x/(sqrt(x)*sqrt(c + d*x)*a + sqrt(x)*sqrt(c + d*x)*b*x**2),x)*a + int 
(x/(sqrt(x)*sqrt(c + d*x)*a + sqrt(x)*sqrt(c + d*x)*b*x**2),x)*b*d + int(1 
/(sqrt(x)*sqrt(c + d*x)*a + sqrt(x)*sqrt(c + d*x)*b*x**2),x)*b*c)/sqrt(e)